DK4141_half 1/20/05 4:56 PM Page 1 Inductors and Transformers for Power Electronics Copyright 2005 by Taylor & Francis Group, LLC DK4141_title 1/20/05 4:55 PM Page 1 Inductors and Transformers for Power Electronics Alex Van den Bossche Ghent University Gent, Belgium Vencislav Cekov Valchev Ghent University Gent, Belgium Boca Raton London New York Singapore A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc. Copyright 2005 by Taylor & Francis Group, LLC The authors try to be accurate and clear, but they cannot guarantee the results or possible interpretations, which might cause direct or indirect injuries, equipment damage, or economic damage by the use of the contents of the book. Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10987654321 International Standard Book Number-10: 1-57444-679-7 (Hardcover) International Standard Book Number-13: 978-1-57444-679-1 (Hardcover) Library of Congress Card Number 2004061860 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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Library of Congress Cataloging-in-Publication Data Bossche, Alex van den. Inductors and tranformers for power electronics / Alex van den Bossche, Vencislav Valchev. p. cm. Includes bibliographical references and index. ISBN 1-57444-679-7 1. Electric inductors. 2. Electric transformers. 3. Power electronics—Equipment and supplies. I. Valchev, Vencislav. II. Title. TK7872.I63B67 2004 621.31'7 dc22 2004061860 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Taylor & Francis Group is the Academic Division of T&F Informa plc. DK4141_Discl Page 1 Friday, February 18, 2005 2:36 PM Copyright 2005 by Taylor & Francis Group, LLC To our children Maxime, Nathan and Laura Cvetelina and Iasen DK4141_C00.fm Page v Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC Preface This book is mainly intended for designers and users of magnetic components in power electronics. It can also be used for didactical purposes. Magnetic components such as inductors and transformers constitute together with the control and the semiconductor components, the main parts in the design of power electronic converters. Some experience teaches that the design of the magnetic parts is still often done by trial and error. This can be explained by a (too) long working-in time for designing inductors and transformers. The design has many aspects, such as the magnetic core and winding, eddy cur- rents, insulation, thermal design, parasitic effects, and measurements. A lot of literature exists concerning those subjects, but the information is spread over many articles and methods. This book is mainly focused on classical methods and uses numerical tools such as finite element methods in the background. We try to give some overview of the basics and technological aspects of the design. In the different chapters we also describe analytical approximations based on known analytical solutions, but tuned by finite elements. In most of the cases, a sufficient accuracy can be obtained and the results are obtained almost instantaneously, even for graphics using many calculation points. A fast approximation method is useful as a first step in the design stage, whereas numerical tools such as finite elements are good in analysis. Specific books on finite elements exist and the description will not be repeated here. Some basic introduction on magnetic principles and materials are given in Chapter 1. Today power electronics use quite a high switching frequency. Simple rules of thumb such as that “the eddy copper current losses are always negligible when the diameter of the wire is smaller than the penetration depth” are not true. However, it is clear that the main cause of the eddy current losses is caused by the presence of high frequency transverse magnetic field compo- nents. This is the base of the fast design method in Chapter 2. The method is further improved using some corrections for other effects and is embedded in a decision flow chart of a design procedure. More insight and better accuracy is provided in the other chapters. We invite the readers to let them guide by the contents of the book to their specific subjects of interest. The chapters in the book are organized in a quite independent way with respective local appendices and references. The general appendices at the end provide information that is not linked to a specific chapter and can be used independently. This work can be seen as complementary information to books on power electronic circuits. Different levels of complexity are proposed depending on the available time, the desired accuracy, and the mathematical level of the designer. DK4141_C00.fm Page vii Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC Acknowledgments We want to thank several institutions, that permitted the research and the achievement of this book: DWTC and FWO – Belgium; NATO Research Program; BOF in Ghent University; Fellowships of scientific exchange between Belgium and Bulgaria; E.E.C. Tempus and Socrates exchange programs. The authors are also grateful to the department heads Prof. Dr. ir Jan Melkebeek of the Ghent University Electrical Energy Laboratory and Prof. Dr. Eng. Dimitar Yudov, who supported us. We want also to acknowledge the companies such as Philips, Tyco, Inverto, Barco, Fabricom in Belgium and Struna in Bulgaria. The opportunity to design for them induced industrial realism. Many collaborators did a wonderful job while reading and giving sugges- tions of improvements and encouragements to the fulfillment of this book. DK4141_C00.fm Page ix Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC About the Authors Alex P. M. Van den Bossche received the M.S. and the Ph.D. degrees from the University of Ghent, Belgium in 1980 and 1990 respectively. He has worked there at the Electrical Energy Laboratory Department, EESA. He has been engaged in research and published articles in the field of electrical drives and power electronics concerning various converter types, drives and various aspects of magnetic components and materials. His interests are also in renewable energy conversion. Since 1993, he has been a full professor at the same university. He is a senior member of the IEEE (M’99S’03). Vencislav V. Valchev received the M.Sc. and Ph.D. degrees in electrical engineering from the Technical University of Varna, Bulgaria in 1987 and 2000, respectively. Since 1988 he has been with the Department of Electronics, Technical University of Varna, where he has been a lecturer. His research interests include power electronics, soft switching converters, resonant con- verters, and magnetic components for power electronics, renewable energy conversion. Dr. Valchev had a cumulated common research period of about four years in the Electric Energy Laboratory research group in Ghent University, Belgium. DK4141_C00.fm Page xi Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC Nomenclature The symbols do mainly follow the standard ISO 31-11 Concerning upper and lower cases we try to keep the following conventions: Voltage and current: Time dependent values of voltage and current are denoted by low cases ( v, i ) RMS values are capitals without index for sinusoidal waveforms. The index rms is mentioned explicitly for RMS values of non-sinusoidal waveforms. Field quantities such as H and B are always written in capital case, the context shows what it is e.g. B p = is the peak value of the induction B ( t ) is the value depending on time. Matrices and vectors are written in bold. Variables are written in italic. Functions, operators, universal constants are non-italic. Complex variables are underlined if confusion is possible. Blanks are used as multiplication. We did split the nomenclature in variables, subscripts, superscripts, constants and frequently used abbreviations. The specific combination of variables with subscripts is defined in the respective chapters at their first appearance. Variables A area [m 2 ] a geometrical dimension [m] B magnetic induction = magnetic flux density [T] b width of the window area, geometrical dimension [m] C coefficient [W/(m 2 K)] c geometrical dimension [m] D duty ratio [] d diameter [m] E electric field [V/m] e dimension [m] F function, factor — f frequency [Hz] = [periods/s] G function — g dimension [m] H magnetic field [A/m] ˆ B DK4141_C00.fm Page xiii Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC xiv Nomenclature i instantaneous current [A] I RMS current (sine wave) [A] k coefficient — k thermal conductivity [W/m ° C] L Inductance [H] L characteristic distance, Chapter 6 [m] l length [m] M total numbers of layers — N number of wires — m layer number — n conductor number in a layer — P power [W] p primary p pressure, Chapter 6 [Pa] q tuning parameter; heat transfer rate [W]; — R resistance; (with index θ : thermal) [ Ω ]:[K/W] = [ ° C; W] r radius [m] S surface [m 2 ] s secondary; distance (with index) —; [m] s Laplace operator —; [m] T period; absolute temperature (with index) [s]; [K] t time; thickness (with index) [s]; [m] V voltage [V] v instantaneous value of the voltage [V] V RMS value of the voltage (sine wave) [V] W area; energy [m 2 ]; [J] w winding width [m] X reactance [ Ω ] x horizontal distance to origin [m] Y admittance [ Ω − 1 ] = [S] y vertical distance to origin [m] z complex distance to origin [m] Z Impedance [ Ω ] α (Alpha) frequency exponent; angle (with index) —; [rad] β (Beta) induction exponent — γ (Gamma) exponent — δ (Delta) penetration depth [m] ε (Epsilon) function; — ε relative number of turns (Chapter 10) — ε emissivity (Chapter 6) — ζ (Zeta) parameter — η (Eta) horizontal filling factor — θ (Theta) angle; temperature [rad][ ° C] κ (Kappa) parameter for the field factor — λ (Lambda) vertical filling factor — µ (Mu) permeability — DK4141_C00.fm Page xiv Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC Nomenclature xv ν (Nu) kinematic viscosity [m 2 /s] ξ (Xi) relative height — ρ (Rho) resistivity [ Ω m] σ (Sigma) conductivity [ Ω − 1 m − 1 ] = [S/m] τ (Tau) time constant [s] Φ (Phi) main flux [Wb] = [Tm − 2 ] ϕ (Phi) angle [rad] χ (Chi) function (influence of penetration depth on dipole effect) — Ψ (Psi) flux linkage [V s] = [T m − 2 ] ψ (Psi) angle [rad] ω (Omega)=2 π f [Hz] = [rad/s] Subscripts 123 number or harmonic A around ( = local) a ambient av average bot bottom (of conductor) c core; curie (temperature), c wide frequency (combined low-high), for coefficients c conductor (for length) cd conduction heat transfer cv convection heat transfer cu copper d differential D Dowell cu copper e effective F from field pattern f finished (area) ff filling factor fe iron, ferrite g gap, graph h thermal h horizontal hf high frequency hy hyperbolic (field type) hs hot spot i,j,k,l,m,n elements of a vector i induced in internal LF low frequency m middle DK4141_C00.fm Page xv Friday, February 18, 2005 3:07 PM Copyright 2005 by Taylor & Francis Group, LLC [...]... so-called parasitic capacitances, and currents in transmission lines This conclusion allows us to use the simplified expression in Equation (1.1) in power electronics magnetic circuit analysis, an approach called the quasi-static approach Copyright 2005 by Taylor & Francis Group, LLC DK4141_C01.fm Page 4 Tuesday, January 18, 2005 11:10 AM 4 Inductors and Transformers for Power Electronics B(t) dl EMF FIGURE... magnetic material The value of mr for air and electrical conductors (e.g., copper, aluminum) is 1 For ferromagnetic materials such as iron, nickel, and cobalt the value of mr is much higher and varies from several hundred to tens of thousands The magnetic flux density B is also called magnetic induction and, for simplicity, in this book we will use the term induction for this magnetic quantity The vector... Supplementary, and Reinforced Insulation 4.4.2 Standard Insulation Distances 4.4.2.1 Clearance 4.4.2.2 Creepage Distance 4.4.3 Electric Strength Tests 4.4.4 Leakage Currents 4.5 Thermal Requirements and Standards 4.5.1 Thermal Evaluation of Insulation Materials and Systems 4.5.2 Requirements and Standards for Inductive (Magnetic) Modules 4.5.3 Standards for Wires Bare Material Diameter Enamel Thickness... chapter gives a brief review of the basic laws, quantities, and units of magnetic theory Magnetic circuits are included together with some examples The analogy between electric and magnetic circuits and quantities is presented Hysteresis and basic properties of ferromagnetic materials are also discussed The models of the ideal transformers and inductors are shown 1.1 Basic Laws of Magnetic Theory The... 2005 by Taylor & Francis Group, LLC S (1.1) DK4141_C01.fm Page 2 Tuesday, January 18, 2005 11:10 AM 2 Inductors and Transformers for Power Electronics Total current i Total density J i1 i2 i3 l FIGURE 1.1 Illustration of Ampere’s law The MMF around a closed loop is equal to the sum of the positive and negative currents passing through the interior of the loop i4 Surface S with area Ac H where H is... Described Methods 8.5.2 Multiple Air Gap Inductors 8.5.3 Avoiding Winding Close to the Air Gap 8.6 Foil Wound Inductors 8.6.1 Foil Inductor—Ideal Case 8.6.2 Single and Multiple Air Gap Design in Foil Inductors 8.6.3 Eddy Current Losses in Foil Windings of Gapped Inductors 8.6.4 Planar Inductors 8.7 Inductor Types Depending on Application 8.7.1 DC Inductors 8.7.2 HF Inductors 8.7.3 Combined DC-HF Inductor... Values of Some Basic Waveforms A.2.1 Discontinuous Waveforms A.2.2 Repeating Line Waveforms A.2.3 Waveforms Consisting of Different Repeating Line Parts A.3 RMS Values of Common Waveforms A.3.1 Sawtooth Wave, Fig A.4 A.3.2 Clipped Sawtooth, Fig A.5 A.3.3 Triangular Waveform, No DC Component, Fig A.6 A.3.4 Triangular Waveform with DC Component, Fig A.7 A.3.5 Clipped Triangular Waveform, Fig A.8 A.3.6 Square... same conditions The atoms of diamagnetic materials Copyright 2005 by Taylor & Francis Group, LLC DK4141_C01.fm Page 6 Tuesday, January 18, 2005 11:10 AM 6 Inductors and Transformers for Power Electronics B Ferromagnetic FIGURE 1.4 Magnetization curves for different types of magnetic materials The scale of the magnetization curve of ferromagnetic materials is much higher Paramagnetic Free air Diamagnetic... recover, but the magnetic moments of the domains will be orientated randomly to each other DK4141_C01.fm Page 8 Tuesday, January 18, 2005 11:10 AM 8 Inductors and Transformers for Power Electronics TABLE 1.1 Curie Temperatures of Various Ferromagnetic Elements and Materials Material Iron Cobalt Nickel Gadolinium Terfenol Alnico Hard ferrites Soft ferrites Amorphous materials Curie temperature, TC, [ºC]... the current i in the coil, and the magnetic field intensity H are zero Increasing the current in the coil results in applying the field with intensity H according the Ampere’s law lc i N FIGURE 1.8 Magnetic core with a coil Copyright 2005 by Taylor & Francis Group, LLC H = Ni /lc DK4141_C01.fm Page 10 Tuesday, January 18, 2005 11:10 AM 10 Inductors and Transformers for Power Electronics B Bsat 3 Br 2 . Inductors and Transformers for Power Electronics Copyright 2005 by Taylor & Francis Group, LLC DK4141_title 1/20/05 4:55 PM Page 1 Inductors and Transformers for. van den. Inductors and tranformers for power electronics / Alex van den Bossche, Vencislav Valchev. p. cm. Includes bibliographical references and index. ISBN